= Coordinate sphere intersection basis
{title2=$[S_I]\cdot[S_J]=1\text{ if }J=I^c,\quad0\text{ otherwise}$}
In the product $(S^{2d})^{2\ell}$, for $d,\ell\ge1$, vary the factors indexed by an $\ell$-element subset $I$ and fix points in the other factors. Orient the resulting <submanifold> $S_I$ in increasing factor order. These <homology classes> form an integral middle-dimensional basis by the <Künneth theorem>. Complementary subsets give a transverse intersection at one point, with positive sign since all factor dimensions are even. For noncomplementary subsets some factor is fixed in both representatives; moving that point makes them disjoint. In particular every basis class has zero <self-intersection number>. Complementary pairs give hyperbolic blocks in the <intersection form>.
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